MathLabs

第4题

确定所有 pairs (n,p)(n,p) 的正整数 使得 p p 是a 素数, n n 不exceeded 2p2p , 且 (p−1)n+1(p-1)^n+1 是divisible 由 np−1 n^{p-1}.
第 4/5 步：Exclude 素数s larger than three
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通俗地说

译文：The first nonzero binomial term has exactly p p 译文：-adic order two.

(p−1)p+1≡p2(modp3)(p-1)^p+1\equiv p^2\pmod {p^3}
详细分析

F或 n=p n=p , binomial expansi上gives (p−1)p+1=(−1+p)p+1=p2+a multiple of p3(p-1)^p+1=(-1+p)^p+1=p^2+\text{a multiple of }p^3. F或 p>3 p>3, divisibility 由 pp−1 p^{p-1} would imply divisibility 由 p3 p^3 译文：, contradiction.